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Multiplier Stabilization Applied to Two-Stage Stochastic Programs

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In many mathematical optimization applications dual variables are an important output of the solving process, due to their role as price signals. When dual solutions are not unique, different solvers or different computers, even different runs in the same computer if the problem is stochastic, often end up with different optimal multipliers. From the perspective of a decision maker, this variability makes the price signals less reliable and, hence, less useful. We address this issue for a particular family of linear and quadratic programs by proposing a solution procedure that, among all possible optimal multipliers, systematically yields the one with the smallest norm. The approach, based on penalization techniques of nonlinear programming, amounts to a regularization in the dual of the original problem. As the penalty parameter tends to zero, convergence of the primal sequence and, more critically, of the dual is shown under natural assumptions. The methodology is illustrated on a battery of two-stage stochastic linear programs

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  • Clara Lage & Claudia Sagastizábal & Mikhail Solodov, 2020. "Multiplier Stabilization Applied to Two-Stage Stochastic Programs," Documents de travail du Centre d'Economie de la Sorbonne 20010, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    • Handle: RePEc:mse:cesdoc:20010
    DOI: 10.1007/s10957-019-01550-7
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    1. Suvrajeet Sen & Yifan Liu, 2016. "Mitigating Uncertainty via Compromise Decisions in Two-Stage Stochastic Linear Programming: Variance Reduction," Operations Research, INFORMS, vol. 64(6), pages 1422-1437, December.
    2. Wim Ackooij & Jérôme Malick, 2016. "Decomposition algorithm for large-scale two-stage unit-commitment," Annals of Operations Research, Springer, vol. 238(1), pages 587-613, March.
    3. Wim Ackooij & Jérôme Malick, 2016. "Decomposition algorithm for large-scale two-stage unit-commitment," Annals of Operations Research, Springer, vol. 238(1), pages 587-613, March.
    4. István Deák, 2006. "Two-stage stochastic problems with correlated normal variables: computational experiences," Annals of Operations Research, Springer, vol. 142(1), pages 79-97, February.
    5. Birge, John R. & Louveaux, Francois V., 1988. "A multicut algorithm for two-stage stochastic linear programs," European Journal of Operational Research, Elsevier, vol. 34(3), pages 384-392, March.
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    More about this item

    Keywords

    Multiplier stability; Dual regularization; Penalty method; Stochastic programming; Two-stage stochastic programming; Empirical approximations;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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